Problem: Given $ \overrightarrow{BA}\perp\overrightarrow{BD}$, $ m \angle ABC = 6x - 6$, and $ m \angle CBD = 5x + 30$, find $m\angle CBD$. $B$ $A$ $D$ $C$
Solution: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Since we are given that $\overrightarrow{BA}\perp\overrightarrow{BD}$ , we know ${m\angle ABD = 90}$ Substitute in the expressions that were given for each measure: $ {6x - 6} + {5x + 30} = {90}$ Combine like terms: $ 11x + 24 = 90$ Subtract $24$ from both sides: $ 11x = 66$ Divide both sides by $11$ to find $x$ $ x = 6$ Substitute $6$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 5({6}) + 30$ Simplify: $ {m\angle CBD = 30 + 30}$ So ${m\angle CBD = 60}$.